Theorem ecovicom 6699

Description: Lemma used to transfer a commutative law via an equivalence relation. (Contributed by Jim Kingdon, 15-Sep-2019.)

Hypotheses

Ref	Expression
ecovicom.1	|- C = ( ( S X. S ) /. .~ )
ecovicom.2	|- ( ( ( x e. S /\ y e. S ) /\ ( z e. S /\ w e. S ) ) -> ( [ <. x , y >. ] .~ .+ [ <. z , w >. ] .~ ) = [ <. D , G >. ] .~ )
ecovicom.3	|- ( ( ( z e. S /\ w e. S ) /\ ( x e. S /\ y e. S ) ) -> ( [ <. z , w >. ] .~ .+ [ <. x , y >. ] .~ ) = [ <. H , J >. ] .~ )
ecovicom.4	|- ( ( ( x e. S /\ y e. S ) /\ ( z e. S /\ w e. S ) ) -> D = H )
ecovicom.5	|- ( ( ( x e. S /\ y e. S ) /\ ( z e. S /\ w e. S ) ) -> G = J )

Assertion

Ref	Expression
ecovicom	|- ( ( A e. C /\ B e. C ) -> ( A .+ B ) = ( B .+ A ) )

Distinct variable groups:   x, y, z, w, A   z, B, w   x, .+ , y, z, w   x, .~ , y, z, w   x, S, y, z, w   z, C, w

Allowed substitution hints:   B( x, y)   C( x, y)   D( x, y, z, w)   G( x, y, z, w)   H( x, y, z, w)   J( x, y, z, w)

Proof of Theorem ecovicom

Step	Hyp	Ref	Expression
1		ecovicom.1	. 2 |- C = ( ( S X. S ) /. .~ )
2		oveq1 5926	. . 3 |- ( [ <. x , y >. ] .~ = A -> ( [ <. x , y >. ] .~ .+ [ <. z , w >. ] .~ ) = ( A .+ [ <. z , w >. ] .~ ) )
3		oveq2 5927	. . 3 |- ( [ <. x , y >. ] .~ = A -> ( [ <. z , w >. ] .~ .+ [ <. x , y >. ] .~ ) = ( [ <. z , w >. ] .~ .+ A ) )
4	2, 3	eqeq12d 2208	. 2 |- ( [ <. x , y >. ] .~ = A -> ( ( [ <. x , y >. ] .~ .+ [ <. z , w >. ] .~ ) = ( [ <. z , w >. ] .~ .+ [ <. x , y >. ] .~ ) <-> ( A .+ [ <. z , w >. ] .~ ) = ( [ <. z , w >. ] .~ .+ A ) ) )
5		oveq2 5927	. . 3 |- ( [ <. z , w >. ] .~ = B -> ( A .+ [ <. z , w >. ] .~ ) = ( A .+ B ) )
6		oveq1 5926	. . 3 |- ( [ <. z , w >. ] .~ = B -> ( [ <. z , w >. ] .~ .+ A ) = ( B .+ A ) )
7	5, 6	eqeq12d 2208	. 2 |- ( [ <. z , w >. ] .~ = B -> ( ( A .+ [ <. z , w >. ] .~ ) = ( [ <. z , w >. ] .~ .+ A ) <-> ( A .+ B ) = ( B .+ A ) ) )
8		ecovicom.4	. . . 4 |- ( ( ( x e. S /\ y e. S ) /\ ( z e. S /\ w e. S ) ) -> D = H )
9		ecovicom.5	. . . 4 |- ( ( ( x e. S /\ y e. S ) /\ ( z e. S /\ w e. S ) ) -> G = J )
10		opeq12 3807	. . . . 5 |- ( ( D = H /\ G = J ) -> <. D , G >. = <. H , J >. )
11	10	eceq1d 6625	. . . 4 |- ( ( D = H /\ G = J ) -> [ <. D , G >. ] .~ = [ <. H , J >. ] .~ )
12	8, 9, 11	syl2anc 411	. . 3 |- ( ( ( x e. S /\ y e. S ) /\ ( z e. S /\ w e. S ) ) -> [ <. D , G >. ] .~ = [ <. H , J >. ] .~ )
13		ecovicom.2	. . 3 |- ( ( ( x e. S /\ y e. S ) /\ ( z e. S /\ w e. S ) ) -> ( [ <. x , y >. ] .~ .+ [ <. z , w >. ] .~ ) = [ <. D , G >. ] .~ )
14		ecovicom.3	. . . 4 |- ( ( ( z e. S /\ w e. S ) /\ ( x e. S /\ y e. S ) ) -> ( [ <. z , w >. ] .~ .+ [ <. x , y >. ] .~ ) = [ <. H , J >. ] .~ )
15	14	ancoms 268	. . 3 |- ( ( ( x e. S /\ y e. S ) /\ ( z e. S /\ w e. S ) ) -> ( [ <. z , w >. ] .~ .+ [ <. x , y >. ] .~ ) = [ <. H , J >. ] .~ )
16	12, 13, 15	3eqtr4d 2236	. 2 |- ( ( ( x e. S /\ y e. S ) /\ ( z e. S /\ w e. S ) ) -> ( [ <. x , y >. ] .~ .+ [ <. z , w >. ] .~ ) = ( [ <. z , w >. ] .~ .+ [ <. x , y >. ] .~ ) )
17	1, 4, 7, 16	2ecoptocl 6679	1 |- ( ( A e. C /\ B e. C ) -> ( A .+ B ) = ( B .+ A ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1364   e. wcel 2164   <.cop 3622   X. cxp 4658  (class class class)co 5919   [cec 6587   /.cqs 6588

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2167  ax-ext 2175  ax-sep 4148  ax-pow 4204  ax-pr 4239

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-un 3158  df-in 3160  df-ss 3167  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-br 4031  df-opab 4092  df-xp 4666  df-cnv 4668  df-dm 4670  df-rn 4671  df-res 4672  df-ima 4673  df-iota 5216  df-fv 5263  df-ov 5922  df-ec 6591  df-qs 6595

This theorem is referenced by:  addcomnqg  7443  mulcomnqg  7445  addcomsrg  7817  mulcomsrg  7819  axmulcom  7933